P-adic Expansion is a Cauchy Sequence in P-adic Norm/Represents a P-adic Number

Theorem

Let $p$ be a prime number.

Let $\ds \sum_{n \mathop = m}^\infty d_n p^n$ be a $p$-adic expansion.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic Numbers.

Then the sequence of partial sums of the series:

$\ds \sum_{n \mathop = m}^\infty d_n p^n$

represents a $p$-adic number of $\struct {\Q_p,\norm {\,\cdot\,}_p}$.

Proof

From P-adic Expansion is a Cauchy Sequence in P-adic Norm, the sequence of partial sums of the series:

$\ds \sum_{n \mathop = m}^\infty d_n p^n$

is a Cauchy Sequence.

Then the sequence of partial sums of the series:

$\ds \sum_{n \mathop = m}^\infty d_n p^n$

is a representative of a $p$-adic number by definition.

$\blacksquare$

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$